Biomedical Engineering Reference
In-Depth Information
Determination of the current is based on the Wentzel, Kramers and Brillouin (WKB)
approximation from which Eq. (1) is obtained.
( )
3/2
⎡
⎤
*
q
ϕ
42
m
2
B
⎢
⎥
J
=
χ
E
exp
−
(1)
FN
FN
3
q
=
E
⎢
⎥
⎣
⎦
where
J
FN
is the current density according to Fowler -Nordheim, χ
FN
is the Fowler
Nordheim constant,
E
is the electric field,
m
*
is the effective mass of the tunneling charge,
=
is a reduced plancks constant,
q
is the electron charge
and φ
B
is the potential barrier
height at the conductor/insulator interface. To check for this current mechanism,
experimental
I
-
V
characteristics are typically plotted as
2
JE
vs 1/
E
, a so-called
Fowler-Nordheim plot. Provided the effective mass of the insulator is known, one can fit the
experimental data to a straight line yielding a value for the barrier height.
ln(
/
)
2.2 Field emission process
Whereas Fowler-Nordheim tunneling implies that carriers are free to move through the
insulator, it cannot be the case where defects or traps are present in an insulator. The traps
restrict the current flow because of a capture and emission process. The two field emission
charge transport process that occur when insulators are sandwiched between metal
electrodes are Poole-Frenkel and Schottky emission process. Thermionic (schottky) emission
assumes that an electron from the contact can be injected into the dielectric once it has
acquired sufficient thermal energy to cross into the maximum potential (resulting from the
superposition of the external and the image-charge potential). If the sample has structural
defects, the defects act as trapping sites for the electrons. Thermally traped charges will
contribute to current density according to Poole-Frenkel emission. They are generally
observed in both organic and inorganic semiconducting materials. Poole-Frenkel effect is
due to thermal excitation of trapped charges via field assisted lowering of trap depth while
Schottky effect is a field lowering of interfacial barrier at the blocking electrode. Expression
for Poole-Frenkel and Schottky effects are given in Eq. (2) and (3) respectively.
1/2
J
=
J
exp[(
β
E kT
) /
]
(2)
PF
PFO
PF
1/2
J
=
J
exp[(
β
E
) /
T
]
(3)
S O
S
J
are pre-exponential factors, β is the Schottky coefficient,
P
β is the Poole-
Frenkel coefficient, and
E
is the electric field. The theoretical values of Schottky and Poole-
Frenkel coefficient are related by Eq.(4):
J
and
SO
PFO
(
)
3
β
=
e
/4
πεε
=
β
/2
(4)
S
0
PF
where
q
is electron charge, ε is relative permittivity, ε
0
permitibity in free space
2.3 Space charge limited current
For structures where carriers can easily enter the insulator and freely move through the
insulator, the resulting charge flow densities are much higher than predicted by Fowler-
Nordheim tunneling and Poole-Frenkel mechanism. The high density of these charged
